8:00 AM Class Group 1
Linear Programming project: Chad Boulier, Daniel Hanley, Sonja Henst,
James Gerosa, Matthew Stone, Amy Tran
Question: Suppose the only foods available in your local supermarket are
meat and potatoes. Suppose each portion of potatoes contains 3 units of
carbohydrates, 4 units of vitamins, 1 unit of protein and costs 75 cents,
while each portion of meat contains 1 unit of carbohydrates, 3 units of
vitamins, 3 units of protein and costs 2 dollars. Suppose also a balanced
diet requires a daily minimum of 8 units of carbohydrates, 19 units of
vitamins and 7 units of protein.
Given the economy, you are greatly concerned about meeting your own
minimum daily requirements while spending as little as possible.
Answer:
Let x be the number of units in potatoes and let y be the number of units
in meat. Let S be the total money spent total on both meat and potatoes
A) 3x + 1y >= 8 ( This equation shows the units of carbohydrates
in potatoes and meat respectively, showing that the sum must be greater
than or equal to 8, as stated in the question.)
B) 4x + 3y >= 19 ( This equation shows the units of vitamins in
potatoes and meat respectively, showing that the sum must be greater than
or equal to 19, as stated in the question.)
C) 1x + 3y >= 7 ( This equation shows the units of protein
potatoes and meat respectively, showing that the sum must be greater than
or equal to 7, as stated in the question.)
D) x >= 0 , y >= 0
E) S = .75x + 2y
Equations: A, B and C are the constraint functions. Where the sum of the
variables (with there respective meanings, must be greater than or equal
to the given value for each constraint function.)
Equation: D, are non-negative conditions because you can not buy a
negative amount of items.
Equation: E is the objective function, where S is in terms the sum of the
total money spent on potatoes (x) and meat (y) for there respective values
of .75 cents and 2 dollars.
Also, the symbols: >= stands for greater than or equal to
And the symbol: (which is not used here) <= would stand for less than or
equal to respectively.
Unlike most linear programming problems this problem is a minimization
problem, where as the optimal solution to the objective function (equation
E) is the smallest possible value that satisfies the constraint functions
(equations A,B,C). The smallest value will be equivalent to the meaning of
"spending as little as possible" as stated in the question.